Quick question about relative velocity in Lorentz Transform

Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.

Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.

Why don't you check this yourself? By definition, the Lorentz transform is constructed so that the world line with ##x' = 0## has ##x = vt##. Can you figure out how to measure the velocity of the spatial origin of the unprimed frame in the primed one?

Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.

Suppose two inertial observers are in relative motion. If they don't measure the same relative velocity, then which one measures the greater velocity? The one moving left? Or the one moving right? Or, the one you choose to call O'? Of the one you choose to call O (without a prime)?

Is it the one whose clock is running slow? But, to each, it's the other's clock that is running slow and whose lengths are contracted.